About this video
The video uses the decibel formula L = 10 log(I / 10^(-12)) to find the sound intensity of a 90-decibel sound. It divides both sides by 10 to isolate the logarithm, rewrites 9 = log(I / 10^(-12)) as 10^9 = I / 10^(-12), and multiplies to get I = 10^(-3), or one thousandth of a watt per square meter.
You'll be able to
- substitute a value into a real-world logarithmic formula.
- isolate a logarithm and rewrite it in exponential form.
- use exponent rules to solve for the unknown and state the answer with units.
Key words
- decibel
- A unit for measuring how loud a sound is (its sound pressure level).
- sound intensity
- How much sound energy passes through an area each second, measured in watts per square meter.
- log base 10
- The exponent you put on 10 to get a number; log(1000) = 3 because 10^3 = 1000.
- exponential form
- Rewriting log_10(A) = c as 10^c = A.
Transcript
- [Instructor] We are told sound pressure level in decibels is calculated using this formula, all right, where I is the sound intensity in watts per meter squared or per square meter. What is the intensity in watts per square meter of a sound measuring 90 decibels.
So pause this video, have a go at that before we do this together. All right, so 90 decibels, so that is going to be right over there. So we have 90 decibels. Well, I won't do the units right now, that might actually make it a little bit confusing, but I'll stick with just the numbers.
So we have 90 decibels is equal to 10 times log base 10 of our sound intensity divided by 10 to the negative 12. All right, so what we wanna do is isolate the logarithm part, and the best way to do that is to divide both sides by 10. So let's do that. So let's divide both sides by 10, and then we are left with nine is equal to log base 10 of our intensity over 10 to the negative 12th power.
Now a logarithm is just another way of saying whatever our base is, if we raise it to that power to the logarithm power, when you take the logarithm, you're essentially getting an exponent. So 10 to the ninth power should be equal to this. So let's write that.
10 to the ninth power should be equal to our intensity over 10 to the negative 12th. So we have 10 to the negative 12th in the denominator. We could multiply both sides by 10 to the negative 12th to solve for I. That cancels with that, and then we get 10 to the negative 12th times 10 to the ninth is 10 to the negative three is equal to I.
And if you wanted to write it without the exponent, you could just write that I is equal to 10 to the negative three. This is 10 to the negative, well, one, two, one, two, three. That is just 1000th. Or another way to think about it is one 10th times one 10th times one 10th, and there you have it.
That is in watts per square meter, and we're done.
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